W.l.o.g., assume that \(S\cap V(AN_n^1)=\lbrace x, y\rbrace \)  and \(S\cap V(AN_n^2)=\lbrace z\rbrace \) . By Lemma REF , Lemma REF  and Menger's theorem [1]}, there are (\(n-1\) ) internally disjoint \((x,y)\) -paths \(P_1, \cdots , P_{n-1}\)  in \(AN_n^1\) . For \(1\le i\le n-1\) , let \(w_i\)  be a neighbour vertex of \(x\)  in \(P_i\) , moreover, let \(\hat{w}_i\)  be the out-neighbour of \(w_i\) .
